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A New Approach to Predicative Set Theory
"... We suggest a new basic framework for the WeylFeferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an a ..."
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We suggest a new basic framework for the WeylFeferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. This idea is implemented using syntactic safety relations between formulas and sets of variables. These safety relations generalize both the notion of domainindependence from database theory, and Godel notion of absoluteness from set theory. The language of PZF is typefree, and it reflects real mathematical practice in making an extensive use of statically defined abstract set terms. Another important feature of PZF is that its underlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation). 1
A Framework for Formalizing Set Theories Based on the Use of Static Set Terms
"... To Boaz Trakhtenbrot: a scientific father, a friend, and a great man. Abstract. We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity ..."
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To Boaz Trakhtenbrot: a scientific father, a friend, and a great man. Abstract. We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity. Like the inconsistent “ideal calculus ” for set theory, it is essentially based on just two settheoretical principles: extensionality and comprehension (to which we add ∈induction and optionally the axiom of choice). Comprehension is formulated as: x ∈{x  ϕ} ↔ϕ, where {x  ϕ} is a legal set term of the theory. In order for {x  ϕ} to be legal, ϕ should be safe with respect to {x}, where safety is a relation between formulas and finite sets of variables. The various systems we consider differ from each other mainly with respect to the safety relations they employ. These relations are all defined purely syntactically (using an induction on the logical structure of formulas). The basic one is based on the safety relation which implicitly underlies commercial query languages for relational database systems (like SQL). Our framework makes it possible to reduce all extensions by definitions to abbreviations. Hence it is very convenient for mechanical manipulations and for interactive theorem proving. It also provides a unified treatment of comprehension axioms and of absoluteness properties of formulas. 1
This document in subdirectoryRS/97/18/ How to Believe a MachineChecked Proof 1
, 909
"... Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS ..."
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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS
A TypeFree Formalization of Mathematics Where Proofs Are Objects
, 1996
"... We present a first order untyped axiomatization of mathematics where proofs are objects in the sense of HeytingKolmogorov functional interpretation. The consistency of this theory is open. ..."
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We present a first order untyped axiomatization of mathematics where proofs are objects in the sense of HeytingKolmogorov functional interpretation. The consistency of this theory is open.
Transitive Closure, Induction, and Logical Frameworks
"... The concept of transitive closure is the key for understanding inductive definitions and inductive reasoning, and so the ability to define the transitive closure of any given relation and make appropriate inferences concerning it is one of the most important challenges that every logical framewor ..."
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The concept of transitive closure is the key for understanding inductive definitions and inductive reasoning, and so the ability to define the transitive closure of any given relation and make appropriate inferences concerning it is one of the most important challenges that every logical framework should meet. We investigate here languages with transitive closure operations and their expressive power. In particular, we show that with the simplest transitive closure operation one can define multiplication from 0, the successor function and addition, but addition is not definable from 0 and the successor function. A stronger version, which binds 4 variables instead of two, allows to define addition as well. We propose also a (necessarily incomplete) corresponding sequent calculus, which suffices for deriving induction as a logical rule. We then show that properly handling all the needed transitive closure operations is problematic for LFstyle logical frameworks. In contrast, this can easily be done in Feferman's FS 0 . Moreover: in this framework the availability of the simplest operation suffices for having all types of inductive definitions (and corresponding inductive principles). 1